Infinitely-Often Autoreducible Sets

A set $A$ is autoreducible if one can compute, for all $x$, the value $A(x)$ by querying $A$ only at places $y \neq x$. Furthermore, $A$ is infinitely-often autoreducible if, for infinitely many $x$, the value $A(x)$ can be computed by querying $A$ only at places $y \neq x$. For all other $x$, the computation outputs a special symbol to signal that the reduction is undefined. It is shown that for polynomial time Turing and truth-table autoreducibility there are $A$, $B$, $C$ in the class EXP of all exponential-time computable sets such that $A$ is not infinitely-often Turing autoreducible, $B$ is Turing autoreducible but not infinitely-often truth-table autoreducible and $C$ is truth-table autoreducible with $g(n)+1$ queries but not infinitely-often Turing autoreducible with $g(n)$ queries. Here $n$ is the length of the input, $g$ is nondecreasing, and there exists a polynomial $p$ such that $p(n)$ bounds both the computation time and the value of $g$ at input of length $n$. Furthermore, connections between notions of infinitely-often autoreducibility and notions of approximability are investigated. The Hausdorff-dimension of the class of sets which are not infinitely-often autoreducible is shown to be $1$.